hw5(1) - T : V → V be lienar. Prove that T 2 = T if and...

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MATH 4377, ADVANCED LINEAR ALGEBRA, SUMMER 2011, HW#5 Due date: Wed., June 16 1. Problem 5 (c), (d) on Page 85. 2. (I wrote this problem on the whiteboard) Let T : M 2 × 2 ( R ) M 2 × 2 ( R ) be a linear transformation and let α = ( 1 0 0 0 ! , 1 1 0 0 ! , 0 1 1 0 ! , 0 0 0 1 !) . Assume that [ T ] α = 1 2 0 0 0 0 1 0 1 1 0 3 0 0 0 1 . For A = a b c d ! M 2 × 2 ( R ), find T ( A ). 3. Problem 2 (b) on Page 96. 4. Problem 3 (a) on Page 96. 5. Problem 4 (a), (b) on Page 97. 6. (see Problem 11 on Page 97). Let V be a vector space and let
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Unformatted text preview: T : V → V be lienar. Prove that T 2 = T if and only if R ( T ) ⊆ N ( T ), where T is the zero transformation, i.e. T ( v ) = 0 for all v ∈ V , and T 2 := T ◦ T is the composition of T and T , i.e. T 2 ( v ) := T ( T ( v )) for all v ∈ V . ( Note : It is a “two-way” statement, you need to prove both directions, i.e prove “= ⇒ ” as well as “ ⇐ =”). 7. Problem 2(a), (e) (f) on Page 106 1...
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This note was uploaded on 02/21/2012 for the course MATH 4377 taught by Professor Staff during the Summer '08 term at University of Houston.

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